Remarks on Commutativity Results for Alternative Rings With [(x^2 y^2+ y^2 x^2),x]=0

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume V, Issue X, October 2020 | ISSN 2454–6186

Remarks on Commutativity Results for Alternative Rings
With [(x^2 y^2+ y^2 x^2),x]=0

Moharram A. Khan¹ ,Abubakar Salisu² and Shu’aibu Salisu^3
1Department of Mathematics and Statistics, Faculty of Natural and applied Sciences, Umaru Musa Yar’adua University, Katsina. Katsina State.
²Science and Technical Education Board Dutse. Jigawa State. Nigeria.
³ Katsina State Science and Technical Education Board. Nigeria..

Abstract: In this article, it is shown that the commutativity of alternative ring satisfying the following properties:
(p_1) [(x^2 y^2+y^2 x^2),x]=0.
(p_2) x(x^2 y^2)=〖(x〗^2 y^2)x.
Keywords: Alternative ring, assosymetric ring, commutator, prime rings.
I. INTRODUCTION
Throughout R represents an alternative ring, C(R) the commutator, A(R) the assosymetric ring. N(R) the set of nilpotent element. An alternative ring R is a ring in which(xx)y=x(xy), y(xx)=(yx)x “for all” “x,y in” R, these equations are known as left and right alternative laws respectively. An assosymetric ring A(R) is one in which (x,y,z)=(p(x),p(y),p(z) ), where p is any permutation of x,y,z∈ R. An associator (x,y,z) we mean by (x,y,z)=(xy)z-x(yz)for all x,y,z ∈ R. A ring R is called a prime if whenever A and B are ideals of R such that AB={0} then either A={0 } or B={0}. If in a ring R, the identity (x,y,x)=0 i.e. (xy)x=x (yx) for all x,y in R holds then R is called flexible. A ring R is said to be m-torsion tree if mx=0 implies x=0, m is any positive number for all x∈R.A non-associative rings R is an additive abelian group in which multiplication is defined, which is distributive over addition on left as well as on right [(x+y)z=xz+yz,z(x+y)=zx+zy,∀ x,y,z ∈R].
Abujabal and Khan [1] proved the commutativity of associative ring satisfies the identity 〖(xy)〗^2=〖xy〗^2 x. Gupta [2] established that a division ring R is commutative if and only if [xy,yx]=0. In addition, Madana and Reddy [3] have established the commutativity of non-associative ring satisfying the identities 〖(xy)〗^2=x^2 y^2 and 〖(xy)〗^2∈Z(R)∀x,y∈R.Further,
Madana Mohana Reddy and Shobha latha.[4] established the commutativity of non-associative primitive rings satisfying the identities:
(〖x(x〗^2+y^2)+(x^2+y^2)x∈Z(R) and x(x〖y)〗^2-(〖xy)〗^2 x∈Z(R).

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